Related papers: Extremal families for the Kruskal--Katona theorem

In \cite[Serra, Vena, Extremal families for the Kruskal-Katona theorem]{sv21}, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in \cite{sv21}…

Given a set $X$ and a sufficiently large integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The…

A fundamental result in extremal set theory is Katona's shadow intersection theorem, which extends the Kruskal-Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of $k$-sets in terms of its size. We…

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal--Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$. The minimum degree…

We prove a minimum degree version of the Kruskal--Katona theorem: given $d\ge 1/4$ and a triple system $F$ on $n$ vertices with minimum degree at least $d\binom n2$, we obtain asymptotically tight lower bounds for the size of its shadow.…

For a set $A\subseteq\left[k\right]^{n}=\left\{ 0,\dots,k-1\right\} ^{n}$, we define the $d$-shadow of $A$ to be the set of points obtained by flipping to zero one of the non-zero coordinates of some point in $A$. Let…

We prove some properties of the Kruskal-Katona function, and apply to the following variation of cross-intersecting antichains. Let $n\ge 4$ be an even integer and $\mathscr{A}$ and $\mathscr{B}$ be two cross-intersecting antichains of…

The Kruskal Katona theorem was proved in the 1960s. In the theorem, we are given an integer $r$ and families of sets $\mathcal{A}\subset \mathbb{N}^{(r)}$ and $\mathcal{B}\subset\mathbb{N}^{(r-1)}$ such that for every $A\in\mathcal{A}$,…

Inspired by the Kruskal-Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of the monoid of increasing functions. One of our main results shows that compressed sets are a…

We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a…

We present a generalization of a version of the Kruskal-Katona theorem due to Lovasz. A shadow of a d-tuple (S_1,...,S_d) in binom{X}{r}^d consists of d-tuples (S_1',...,S_d') in binom{X}{r-1}^d obtained by removing one element from each of…

Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erd\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum…

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal…

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\mathcal{A}, \mathcal{B}…

Given a family $\mathcal F\subset 2^{[n]}$, its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of $k$-uniform intersecting families. In…

The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollob\'as and Eccles, called the partial shadow problem. The problem asks…

For a family $\mathcal{F}$ of words of length $n$ drawn from an alphabet $A=[r]=\{1,\dots,r\}$, Danh and Daykin defined the deletion shadow $\Delta \mathcal{F}$ as the family containing all words that can be made by deleting one letter of a…

Let $n > k > t \geq j \geq 1$ be integers. Let $X$ be an $n$-element set, ${X\choose k}$ the collection of its $k$-subsets. A family $\mathcal F \subset {X\choose k}$ is called $t$-intersecting if $|F \cap F'| \geq t$ for all $F, F' \in…

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain. Erdos extended this theorem to determine the largest family…

The study of intersection problems on families of sets is one of the most important topics in extremal combinatorics. As we all know, the extremal problems involving certain intersection constraints are equivalent to that with the union…